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A subset A of a Banach space is called Banach-Saks when every sequence in A has a Cesaro convergent subsequence. Our interest here focuses on the following problem: is the convex hull of a Banach-Saks set again Banach-Saks? By means of a combinatorial argument, we show that in general the answer is negative. However, sufficient conditions are given in order to obtain a positive result. (C) 2013 Elsevier Inc. All rights reserved.